Duopoly prices under congested access*

Abstract. Consider two firms, at different locations, supplying a homogenous good at constant marginal production cost. Consumers incur travel costs to the firm for each unit purchased, and the travel costs increase with the amount of travel to each firm (congestion). When all traffic and all congestion are generated by travel to a duopolist, both the Nash–Bertrand equilibrium prices and the Nash–Cournot equilibrium prices exceed the sum of the marginal production cost and the marginal external travel cost. However, when the road is shared by travelers to the duopolists' facilities and travelers in competitive markets, the Nash–Bertrand duopoly price equals the competitive price and the Nash–Cournot price contains a markup.     

MARKET CAPTURE BY TWO COMPETITORS: THE PREEMPTIVE LOCATION PROBLEM*

ABSTRACT. We consider a location and allocation game for two competitor firms, A and B, that each seek to locate p facilities in a network. A market is captured by a particular firm if that market's closest facility belongs to that firm rather than a competitor. The question is as follows: Firm A wants to locate its p facilities so that B, which enters also with p facilities after Firm A has located its facilities, will capture the minimum market value possible. That is, Firm A wishes to preempt Firm B in its bid to capture market share to the maximum extent possible. A model is presented that addresses this issue, together with solution methods and computing times.     

On the Finite Optimality Set of the Vector Assignment p‐Median Problem

The vector assignment p‐median problem (VAPMP) is one of the first discrete location problems to account for the service of a demand by multiple facilities, and has been used to model a variety of location problems in addressing issues such as system vulnerability and reliability. Specifically, it involves the location of a fixed number of facilities when the assumption is that each demand point is served a certain fraction of the time by its closest facility, a certain fraction of the time by its second closest facility, and so on. The assignment vector represents the fraction of the time a facility of a given closeness order serves a specific demand point. Weaver and Church showed that when the fractions of assignment to closer facilities are greater than more distant facilities, an optimal all‐node solution always exists. However, the general form of the VAPMP does not have this property. Hooker and Garfinkel provided a counterexample of this property for the nonmonotonic VAPMP. However, they do not conjecture as to what a finite set may be in general. The question of whether there exists a finite set of locations that contains an optimal solution has remained open to conjecture. In this article, we prove that a finite optimality set for the VAPMP consisting of “equidistant points” does exist. We also show a stronger result when the underlying network is a tree graph.     

HOTELLING'S “MAIN STREET” WITH MORE THAN TWO COMPETITORS*

ABSTRACT. I analyze oligopolistic competition among three or more firms located on Hotelling's (1929) Main Street and show that in contrast with Hotelling's duopoly, the symmetric locational structure supports a noncooperative equilibrium in prices. However, in a two-stage game of location choice in the first stage, and price choice in the second stage, there exists no subgame-perfect equilibrium where the whole market is served. This is because, starting from any locational pattern, firms have incentives to move toward the central firm. This strong version of the Principle of Minimum Differentiation destroys the possibility of a locational equilibrium. The results are a direct consequence of the existence of boundaries in the space of location. The sharp difference between these results and those of the standard circular model (whose product space lacks boundaries) shows that the general use of the circular model as an approximation to the line interval model may be unwarranted.     

GENERAL EQUILIBRIUM OF SPATIAL PRODUCT AND LABOR MARKETS*

ABSTRACT A simple general equilibrium model relates spatial product markets and spatial labor markets. The firm is treated as being a spatial monopolist or as a Löschian competitor in the output market and as a spatial monopsonist in the labor market. Derived free spatial demand and free regional labor supply are defined, and their properties examined. The model provides the framework for analyzing the impact of a technological improvement in labor productivity on the structure of the spatial markets. The impact of entry on spatial labor supply is an important determinant of whether or not entry lowers wages and raises output prices. Unlike the spaceless competitive paradigm, zero-profit long-run equilibrium can occur in a space economy under conditions of increasing returns to scale.     

A SPATIAL COMPETITION APPROACH TO CENTRAL PLACE THEORY: SOME BASIC PRINCIPLES*

ABSTRACT. This paper investigates the potential offered by the model of spatial competition for the study of central place theory. We consider n firms selling m substitutable or complementary goods to a continuum of consumers evenly distributed along a linear segment. Consumers have the same income and the same utility function which is quadratic in the goods supplied by the firms and linear in the numeraire. The main results are as follows. (1) In any location equilibrium in which all goods are consumed everywhere, each good supplied by a single firm is sold at the market center. In Christaller's terminology, this means that when the exhaustive principle holds in equilibrium, highest-order goods are made available at the center. (2) When all goods (excluding the numéraire) are complements to each other and each good is sold by a single firm, there always exists an equilibrium in which all the firms locate coincidentally. (3) If the stores selling a given good are under the control of a single owner then, in any equilibrium for which the exhaustive principle holds, the stores are located in a way such that the total transport cost (borne by consumers) is minimized.     

Multiple Facilities Location in the Plane Using the Gravity Model

Two problems are considered in this article. Both problems seek the location of p facilities. The first problem is the p median where the total distance traveled by customers is minimized. The second problem focuses on equalizing demand across facilities by minimizing the variance of total demand attracted to each facility. These models are unique in that the gravity rule is used for the allocation of demand among facilities rather than assuming that each customer selects the closest facility. In addition, we also consider a multiobjective approach, which combines the two objectives. We propose heuristic solution procedures for the problem in the plane. Extensive computational results are presented.     

Surviving in a Competitive Spatial Market: The Threshold Capture Model

Many facility location decision models ignore the fact that for a facility to survive it needs a minimum demand level to cover costs. In this paper we present a decision model for a firm that wishes to enter a spatial market where there are several competitors already located. This market is such that for each outlet there is a demand threshold level that has to be achieved in order to survive. The firm wishes to know where to locate its outlets so as to maximize its market share taking into account the threshold level. It may happen that due to this new entrance, some competitors will not be able to meet the threshold and therefore will disappear. A formulation is presented together with a heuristic solution method and computational experience.     

Modeling Geographic Ferrous Scrap Markets: Regional Prices and Interregional Transactions in the United States*

Abstract. The U.S. Regional Ferrous Scrap Model analyzes spatial variations in prices for two grades of ferrous scrap using a logistic model of choice under differentiated products. The model uses a computer‐generated equilibrium framework to solve for prices that support the observed spatial distribution of supply and demand quantities. This paper presents the model's formal structure and its solution algorithm. The model specification is highly disaggregated with 1,212 supply and 240 demand regions. Characteristics of the equilibrium solution are described for prices and interregional flows. Sensitivity of equilibrium values to changes in model parameters is reported.     

The Location Invariance Theorem in Weberian Space Under Uncertainty

This paper examines the validity of the location invariance theorem in Weberian space under various types of uncertainty. The main results are: Given that the firm's location is constrained to remain at a specified distance from the output market, the optimal location is invariant to any change in product demand if and only if the production function is homothetic for a firm facing demand price uncertainty, or if the production function is homothetic and both inputs are risk-neutral for a firm facing technological uncertainty. Alternatively, given that the distance from the firm's location to the output market is a variable, location invariance occurs for a firm facing demand price uncertainty if the production function is linear homogeneous. In the presence of input price uncertainty the optimal location always varies with a change in product demand. The results can include those previously obtained for linear stochastic location models as special cases and some are new contributions to the literature.     

PRODUCT DIFFERENTIATION,PRICING POLICY AND EQUILIBRIUM*

ABSTRACT. This paper studies the price-location equilibrium of duopolists supplying differentiated goods and competing in a spatial market with elastic demand. We show that a price-location equilibrium exists under all three pricing policies traditionally considered by the literature: f.o.b. mill, uniform delivered, and spatially discriminatory pricing. We also show that firms always cluster at the market center. The second part of the paper studies the endogenous choice of pricing policy. A surprising feature of the resulting equilibrium is asymmetry. The greater the extent to which the goods are substitutes, the more likely is it that one firm will choose f.o.b. pricing and the other price discrimination. Finally, the welfare consequences of the analysis show some interesting trade-offs.     

Long–Run Industry Equilibrium Under Demand Uncertainty: The Locational Effect of Business Taxes

This paper departs from previous work on the locational effects of business taxes under uncertainty by considering the attainment of long–run industry equilibrium. A general m input, planar–space production location model is adopted, and each individual firm is assumed to face random industry demand. I provide some interesting results which are contrary to those proposed by previous studies in the single–firm context and the relative impact of various business taxes.     

SPATIAL COMPETITION,PRICING, AND MARKET POWER IN TRANSPORTATION: A DOMINANT FIRM MODEL*

ABSTRACT Transport firms compete over space. We develop a dominant firm model of transport services wherein one firm (the railroad) has market power, but competes in space with a competitive alternative (truck‐barge). When constrained, the dominant firm prices to “beat the competition”, which impedes efficiency when demand has some elasticity. We extend the basic model in a number of directions that include the relationship between monopoly prices and the generalized concavity of the shipper demand functions, the effects of multiple terminal markets, the role of joint production (fronthaul–backhaul markets), and the effects of capacity constraints.     

LOCATION OF PUBLIC FACILITIES WITH SPILLOVER EFFECTS: VARIABLE LOCATION AND PARAMETRIC SCALE*

ABSTRACT This paper studies the location of public facilities of two neighboring local governments which consider not only the influence of the land market but also the spillover effects that each jurisdiction may have on the other. We obtain the following results: (1) in most cases, one of the cities behaves as an isolated city in choosing the facility location while the other enjoys the spillover effect as a free rider; (2) we also find that the equilibrium location in the two noncooperative city case is not socially optimal except for a special case.     

A MANAGERIAL MODEL OF THE EFFECT OF PRODUCT MARKET STRUCTURE ON FIRM LOCATION*

ABSTRACT. This paper develops a model of firm location choice based on managerial theory of discretionary behavior. Specifically, it is assumed that the management of the firm maximizes a utility function which incorporates profits and location-specific amenities. As the firm moves from one prospective location to another, it faces a profit-amenity constraint imposed by market conditions. The optimal location decision is derived by maximizing the utility function subject to this market-imposed constraint. After examining the properties of the optimal solution, the impact of various changes in product market structure (including changes from a contestible markets perspective) on the location decision is investigated. A major finding is that the impact of a change in market structure depends upon the nature of the structure change and upon the substitution and income effects induced by the structural change.     

Determining Market Areas Captured by Competitive Facilities: A Continuous Equilibrium Modeling Approach

Many existing models concerning locations and market areas of competitive facilities assume that customers patronize a facility based on distance to that facility, or perhaps on a function of distances between the customer and the different facilities available. Customers are generally assumed to be located at certain discrete demand points in a two-dimensional space, or continuously distributed over a one-dimensional line segment. In this paper these assumptions are relaxed by employment of a continuum optimization model to characterize the equilibrium choice behavior of customers for a given set of competitive facilities over a heterogeneous two-dimensional space. Customers are assumed to be scattered continuously over the space and each customer is assumed to choose a facility based on both congested travel time to the facility and on the attributes of the facility. The model is formulated as a calculus of variations problem and its optimality conditions are shown to be equivalent to the spatial customer-choice equilibrium conditions. An efficient numerical method using finite element technique is proposed and illustrated with a numerical example.     

THE EFFECTS OF ZONING IN SPATIAL COMPETITION

ABSTRACT This paper considers a location model to illustrate the effect of zoning on competition. A planner is in charge of designing a city in a circular space where firms and consumers are located on different sides. With this type of market configuration, equilibrium in location under concave transport costs is proved. Then, a welfare function with different weights attached to consumer and firm surpluses is introduced to highlight zoning regulation as an influential competition policy tool. Depending on the regulator's political profiles and the demand, it is shown that zoning can lead to strong, weak, or moderate competition.     

The Optimal Nodal Location of Public Facilities With Price-Sensitive Demand

The purpose of this paper is to present some models for the location of public facilities in nodal networks that explicitly maximize social welfare by accounting for price-elastic demand functions. The models presented here are general; yet they are mathematically equivalent to the plant location problem and are therefore amenable to solution procedures developed for the plant location problem. The models presented here distinguish between two institutional environments that reflect the degree of power of the consumer to choose which facility to patronize. If consumers can be assigned arbitrarily to facilities and can be denied service, then the environment is one of public fiat. If consumers must be served at the facility of their choice, then a “serve-allcomers” environment exists. Separate models for each environment are specified, and the relationship between optimal assignments and pricing policies is developed.     

SPATIAL COMPETITION À LA COURNOT: PRICE DISCRIMINATION BY QUANTITY-SETTING OLIGOPOLISTS*

In this paper, we consider oligopolistic competition in a spatial model when firms take care of goods' delivery and discriminate among consumers. Firms compete by setting quantity schedules independently over space. We show that under general conditions a Nash equilibrium in this game exists and is unique. In equilibrium, firms’ markets overlap, a feature which accords with intuition and empirical observations. Over the interval between two firms, the equilibrium spatial price schedule is quasi-concave (quasi-convex) when transport costs are concave (convex). With linear transport costs, the model predicts uniform delivered pricing. Uniform pricing could moreover be obtained by a combination of increasing returns to volume in transportation together with concavity of unit transport costs in distance.     

THE OPTIMAL LOCATIONS OF BRANCH FACILITIES AND MAIN FACILITIES WITH CONSUMER SEARCH*

ABSTRACT. This paper examines the socially optimal locations of branch facilities (or small stores) and main facilities (or large stores) on a finite linear market that is uniformly populated from position 0 to position 1. Each consumer has a probability w of finding the desired service (or product) at a branch facility, and a probability 1 of finding the desired service (or product) at a main facility. Two types of consumer search are considered: phone search and visit search. Different assumptions are made about the numbers of branch facilities and main facilities (each involving one or two facilities of each type). Under visit search, the socially optimal locations of branch facilities tend to be closer to main facilities than under phone search, and this tendency is more pronounced for smaller values of w.